The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

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A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and I have only tested it on the current versions of Chrome and Firefox (it should work in most browsers but I make no guarantees). Also, the MIT math web server has recently developed the habit of occasionally throwing 403 errors when you try to load a page — if this happens just hit reload.

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A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and it has only been tested on the current versions of Chrome and Firefox.

== Errata ==

== Errata ==

Revision as of 09:23, 12 October 2013

This is the home page for the Polymath8 project "bounded gaps between primes".

Contents

World records

[math]H[/math] is a quantity such that there are infinitely many pairs of consecutive primes of distance at most [math]H[/math] apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).

[math]k_0[/math] is a quantity such that every admissible [math]k_0[/math]-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in [math]k_0[/math] lead to improvements in [math]H[/math]. (The relationship is roughly of the form [math]H \sim k_0 \log k_0[/math]; see the page on finding narrow admissible tuples.)

[math]\varpi[/math] is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in [math]\varpi[/math] lead to improvements in [math]k_0[/math], as described in the page on Dickson-Hardy-Littlewood theorems. In more recent work, the single parameter [math]\varpi[/math] is replaced by a pair [math](\varpi,\delta)[/math] (in previous work we had [math]\delta=\varpi[/math]). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on distribution of primes in smooth moduli.

In this table, infinitesimal losses in [math]\delta,\varpi[/math] are ignored.

[math]k_0[/math] bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.

Systematic tables of tuples of small length have been set up here and here (update: As of June 27 these tables have been merged and uploaded to an online database of current bounds on [math]H(k)[/math] for [math]k[/math] up to 5000).

Tuples applet

Here is a small javascript applet that illustrates the process of sieving for an admissible 632-tuple of diameter 4680 (the sieved residue classes match the example in the paper when translated to [0,4680]).

The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and it has only been tested on the current versions of Chrome and Firefox.

Errata

Page 5: In the first display, [math]\mathcal{E}[/math] should be multiplied by [math]\mathcal{L}^{2k_0+2l_0}[/math], because [math]\lambda(n)^2[/math] in (2.2) can be that large, cf. (2.4).

Page 14: In the final display, the constraint [math](n,d_1=1[/math] should be [math](n,d_1)=1[/math].

Page 35: In the display after (10.5), the subscript on [math]{\mathcal J}_i[/math] should be deleted.

Page 36: In the third display, a factor of [math]\tau(q_0r)^{O(1)}[/math] may be needed on the right-hand side (but is ultimately harmless).

Page 38: In the display after (10.14), [math]\xi(r,a;q_1,b_1;q_2,b_2;n,k)[/math] should be [math]\xi(r,a;k;q_1,b_1;q_2,b_2;n)[/math].

Page 42: In (12.3), [math]B[/math] should probably be 2.

Page 47: In the third display after (13.13), the condition [math]l \in {\mathcal I}_i(h)[/math] should be [math]l \in {\mathcal I}_i(sh)[/math].

Page 49: In the top line, a comma in [math](h_1,h_2;,n_1,n_2)[/math] should be deleted.

Page 51: In the penultimate display, one of the two consecutive commas should be deleted.

Page 54: Three displays before (14.17), [math]\bar{r_2}(m_1+m_2)q[/math] should be [math]\bar{r_2}(m_1+m_2)/q[/math].

Errata for Motohashi-Pintz's "A smoothed GPY sieve", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper.

Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of [math](\log \frac{R}{|D|})^{2\ell+1}[/math], (4.15) contains instead a factor of [math](\log \frac{R/w}{|K|})^{2\ell+1}[/math] which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of [math]\exp(-k\omega/3)[/math] in (4.15) does not seem to be available for estimating the second sum in (5.14).

Page 7: In (2.39), the exponent of [math]3a/2[/math] should instead be [math]-5a/2[/math] (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.

Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

[R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. MathSciNetArticle